Struct for mark distributions

[JoseKling]

The current implementations only account for mark distributions that are independent from the event times. This is mainly focused on using Distributions.jl as mark distributions.
There are some obvious limitations to that and it is also a bit cumbersome at times. Some issues:

• It is not straightforward to use a PoissonProcess with a mark distribution that depends on time or the history
• Using the Dirac(nothing) for unmarked processes is annoying and not intuitive. It has no logdensityof method, which breaks some methods.

My idea is to add a AbstractMarkDistribution that can be easily extended and implements the necessary interface independently of the process. We could have things like

abstract type AbstractMarkDistribution end

sample_mark(md::AbstractMarkDistribution, h, t) = rand(md.mark_dist)
DensityInterface.densityof(md::FixedDistribution, h, t, m) = densityof(mark_distribution(md.mark_dist, h, t), m)

# Replaces `Dirac(nothing)` 
struct NoMarks <: AbstractMarkDistribution end

mark_distribution(md::NoMarks, t , h) = Dirac(nothing)
DensityInterface.densityof(md::NoMarks, h, t, nothing) = 1.0 # Fix density issue

# Works with distributions from Distribution.jl
struct FixedDistribution{D} <: AbstractMarkDistribution 
    mark_dist::D
end

mark_distribution(m::FixedDistribution, h, t) = m.mark_dist()

# Easily extend to other cases
struct NonStationaryNormalDistribution{R<:Real} <: AbstractMarkDistribution
    a::R
    b::R
end
mark_distribution(md::NonStationaryNormalDistribution, t , h) = Normal( a * t + b)

would require a bit more thought on the interface, though.

[JoseKling]

@rsenne you think this makes sense? If so, I think I would work on this and then resume the multivariate Hawkes PR.

[rsenne]

I like this idea. I'm curious if we can maybe play around with the FixedDistirbution part though. I don't love needing a wrapper as I imagine for most use cases, a user just wants a standard Distributions.jl object.

So iterating on your idea:

abstract type AbstractMarkDistribution end

mark_distribution(md::AbstractMarkDistribution, h, t) = md

sample_mark(md, h, t) = rand(mark_distribution(md, h, t))

DensityInterface.densityof(md, h, t, m) =
    densityof(mark_distribution(md, h, t), m)

# support Distributions directly
mark_distribution(d::Distribution, h, t) = d

# now we can make an explicit NoMarks()
struct NoMarks <: AbstractMarkDistribution end

mark_distribution(::NoMarks, h, t) = Dirac(nothing)

sample_mark(::NoMarks, h, t) = nothing

DensityInterface.densityof(::NoMarks, h, t, ::Nothing) = 1.0
DensityInterface.densityof(::NoMarks, h, t, m) = 0.0

What do you think?

[JoseKling]

Yes, this is simpler and I think will work great. I will start with this and see if there is anything we are not considering.